Contents

cohomology

# Contents

## Idea

The equivariant cohomology-generalization of de Rham cohomology.

## Properties

Throughout we consider the following setup:

###### Definition

(smooth manifold with smooth action of a Lie group)

Let

1. $X$ be a smooth manifold with de Rham algebra denoted $\big( \Omega^\bullet(X), d_{dR} \big)$,

2. $G$ a Lie group with Lie algebra denoted $\big(\mathfrak{g}, [-,-]\big)$,

3. $X \times G \overset{\rho}{\longrightarrow} X$ a smooth action of $G$ on $X$.

Often this data is called a smooth $G$-manifold $X$, or similar.

### Models

Given a smooth $G$-manifold $X$ (Def. ) various dg-algebras are used to model the corresponding $G$-equivariant de Rham cohomology $X$, known as

#### The Weil model

Let $\big( W(\mathfrak{g}), d_W \big)$ denote the Weil algebra of $\mathfrak{g}$. If $\{t_a\}$ is a linear basis for $\mathfrak{g}$

(1)$\mathfrak{g} \;=\; span\big( \{t_a\} \big)$

with induced structure constants for the Lie bracket being $\big\{f_{b c}^a\big\}_{a,b,c}$

(2)$[t_a, t_b] \;=\; f_{a b}^c t^b \wedge t^c$

(using the Einstein summation convention throughout)

then the Weil algebra is the dgc-algebra explicitly given by generators and relations as follows:

(3)W(\mathfrak{g}) \;\coloneqq\; \mathbb{R}\big[ \{ \underset { deg = 1 } { \underbrace{ t^a } } \}_a, \{ \underset { deg = 2 } { \underbrace{ r^a } } \}_a \big] \Big/ \left( \begin{aligned} d_W \, t^a & = - \tfrac{1}{2}f^a_{b c} t^b \wedge t^c + r^a \\ d_W \, r^a & = f^a_{b c} t^b \wedge r^c \end{aligned} \right)

Now consider the tensor product of dgc-algebras of the de Rham algebra of $X$ with the Weil algebra of $\mathfrak{g}$

(4)$\Big( \Omega^\bullet \big( X \big) \otimes W(\mathfrak{g}), \, d_{dR} + d_W \Big) \,.$

On this consider the following joint Cartan calculus operations: for each basis element $t_a$ (1) a graded derivation of degree -1 (contraction)

(5)$\iota_a \;\colon\; \Big( \Omega \big( X \big) \otimes W(\mathfrak{g}) \Big)^{\bullet} \longrightarrow \Big( \Omega \big( X \big) \otimes W(\mathfrak{g}) \Big)^{\bullet - 1 }$

and a graded derivation of degree 0 (generalized Lie derivative)

(6)$\mathcal{L}_a \;\colon\; \Big( \Omega \big( X \big) \otimes W(\mathfrak{g}) \Big)^\bullet \longrightarrow \Big( \Omega \big( X \big) \otimes W(\mathfrak{g}) \Big)^{\bullet}$

defined on $\omega \in \Omega^\bullet(X)$ any differential form and $t^a, r^a$ as in (3) as follows

(7)\iota_a \;\colon\; \left\{ \begin{aligned} \omega & \mapsto \iota_{v^a} \omega \\ t^b &\mapsto \delta^a_b \\ r^b & \mapsto 0 \end{aligned} \right.

and

(8)\mathcal{L}_a \;\colon\; \left\{ \begin{aligned} \omega & \mapsto \mathcal{L}_{v^a} \omega \\ t^b &\mapsto f_{a c}^b t^c \\ r^b & \mapsto f_{a c}^b r^c \end{aligned} \right.

where

$v^a \;\colon\; X \overset{ \big( (e,t_a), 0 \big) }{\hookrightarrow} T G \times T X \simeq T ( G \times X ) \overset{ d \rho } {\longrightarrow} T X$

is the vector field on $X$ which is the derivative of the action $\rho$ of $G$ along the Lie algebra-element $t_a \in \mathfrak{g} \simeq T_e G$,

and where $\iota_{v^a}$ is ordinary contraction of vector fields into differential forms and $\mathcal{L}_{v^a} = [d_{dR}, \iota-{v^a}]$ is Lie derivative of differential forms.

With this one defines the sub-chain complex of horizontal differential forms as the joint kernel of the contraction operators (5)

(9)$\Big( \Omega^\bullet \big( X \big) \otimes W(\mathfrak{g}) \Big)_{hor} \overset{ ker\big( \{\iota_a\}_a \big) }{\hookrightarrow} \Big( \Omega^\bullet \big( X \big) \otimes W(\mathfrak{g}) \Big)$

(this subspace need not be preserved by the differential, but the following further subspace is)

and the further sub-dgc-algebra of basic differential forms, which are in addition in the kernel of the Lie derivatives (6)

(10)$\Big( \Omega^\bullet \big( X \big) \otimes W(\mathfrak{g}) \Big)_{basic} \overset{ ker\big( \{\mathcal{L}_a\}_a \big) }{\hookrightarrow} \Big( \Omega^\bullet \big( X \big) \otimes W(\mathfrak{g}) \Big)_{hor} \overset{ ker\big( \{\iota_a\}_a \big) }{\hookrightarrow} \Big( \Omega^\bullet \big( X \big) \otimes W(\mathfrak{g}) \Big)$

Since the differential $d_{dR} + d_W$ (4) graded-commutes with $\iota_a$ to $\mathcal{L}_a$ (by definition and by Cartan's magic formula) and hence graded-commutes with the Lie derivative $\mathcal{L}_a$ itself, it restricts to this joint kernel, thus defining a sub-dgc-algebra (just no longer semi-free, in generaL)

(11)$\Big( \Big( \Omega^\bullet \big( X \big) \otimes W(\mathfrak{g}) \Big)_{basic} , d_{dR} + d_W \Big)$

This dgc-algebra is called the Weil model for $G$-equivariant de Rham cohomology of $X$.

#### The Cartan model

The Cartan model follows from the Weil model above by algebraically solving the horizontality constraint (9). This we discuss first below. Then we describe the resulting dgc-algebra further below.

Reviews include (Mathai-Quillen 86, Sec. 5, Kalkman 93, section 2.2)

##### Via horizontal projection of the Weil model

The Cartan model arises form the Weil model above by the observation that the first of the two constraints defining basic differential forms (10), namely the constrain for horizontal differential forms (9), may be uniformly solved:

###### Lemma

(projection operator onto horizontal differential forms)

Consider the normal ordered exponential of minus the sum of the contraction derivations (5) followed by wedge product with the corresponding degree-1 generator (3)

(12)$: \exp \big( - \theta^a \iota_a \big) : \;=\; 1 - \underset{a}{\sum} \theta^a \iota_a + \tfrac{1}{2} \underset{a,b}{\sum} \theta^a \theta^b \iota_b \iota_a - \cdots \;\;\colon\;\; \Omega^\bullet \big( X \big) \otimes W(\mathfrak{g}) \longrightarrow \Omega^\bullet \big( X \big) \otimes W(\mathfrak{g})$

We have:

1. This is the projection operator onto the sub-space of horizontal differential forms (9).

2. The restriction of this projector to $\Omega^\bullet\big(X \big)$ is a graded algebra-isomorphism onto the horizontal forms in $CE(\mathfrak{g}) \otimes \Omega^\bullet\big(X \big)$

$\Omega^\bullet\big(X \big) \underoverset{\simeq}{ :\exp\big( - \theta^a \iota_a \big): }{\longrightarrow} \big( CE(\mathfrak{g}) \otimes \Omega^\bullet(X) \big)_{hor}$
3. Hence the further tensor product with $\mathbb{R}\big[ \{r^a\}_a \big]$ is an algebra isomorphism onto the full subspace of horizontal differential forms (9)

$\mathbb{R}\big[ \{r^a\}_a \big] \otimes \Omega^\bullet\big(X \big) \underoverset{\simeq}{ :\exp\big( - \theta^a \iota_a \big): }{ \longrightarrow } \big( CE(\mathfrak{g}) \otimes \Omega^\bullet(X) \big)_{hor}$
4. The operator commutes with the Lie derivative (6) and hence restricts to an isomorphism onto the sub-dgc-algebra of basic differential forms (10)

(13)$\big( \mathbb{R}\big[ \{r^a\}_a \big] \otimes \Omega^\bullet\big(X \big) \big)^G \underoverset{\simeq}{ \;\;\; :\exp\big( - \theta^a \iota_a \big): \;\;\; }{ \longrightarrow } \big( CE(\mathfrak{g}) \otimes \Omega^\bullet(X) \big)_{bas}$
5. The inverse of (13)

(14)$\big( \mathbb{R}\big[ \{r^a\}_a \big] \otimes \Omega^\bullet\big(X \big) \big)^G \underoverset{\simeq}{ \;\;\; \epsilon \;\;\; }{ \longleftarrow } \big( CE(\mathfrak{g}) \otimes \Omega^\bullet(X) \big)_{bas}$

is the algebra homomorphism given setting all generaotors $\theta^a$ in (3) to zero

(15)$\epsilon \;\colon\; \left\{ \array{ \omega & \mapsto \omega \\ \theta^a & \mapsto 0 \\ r^a & \mapsto r^a } \right.$
6. The induced differential on the left, which hence makes $: \exp\big( - \theta^a \iota_a\big) :$ a dgc-algebra-isomorphism and hence in particular a quasi-isomorphism is

(16)$\epsilon \circ \big( d_{dR} + d_W\big) \circ : \exp\big( -\theta^a \iota_a \big) : \;=\; d_{dR} + r^a \iota_{v^a} \,.$

This is the Mathai-Quillen isomorphism (Mathai-Quillen 86, around (5.9)).

###### Proof

Observe that the operator (12) is equal to the product

$: \exp \big( - \theta^a \iota_a \big) : \;=\; \big( id - \theta^1 \iota_1 \big) \big( id - \theta^2 \iota_2 \big) \cdots \big( id - \theta^{dim(\mathfrak{g})} \iota_{dim(\mathfrak{g})} \big) \,.$

Here all factors commute with each other, and each factor is itself a projection operator, with image the kernel of the corresponding single contraction operator, e.g.

$im \big( 1 - \theta^1 \iota_1 \big) \;\simeq\; ker\big( \iota_1\big)$

etc.

Hence the joint image is the joint kernel of the contraction operators.

It is clear by inspection that $\epsilon$ in (14) is a linear inverse to $: \exp\big( - \theta^a \iota_a\big) :$. Therefore, since $\epsilon$ is manifestly an algebra homomorphism, so is $: \exp\big( - \theta^a \iota_a\big) :$.

This implies that the induced differential (16) is a graded derivation and hence that it may be identified by its action on generators. Direct inspection indeed yields

for all generators $r^a$

\begin{aligned} \epsilon \circ \big( d_{dR} + d_W\big) \circ : \exp\big( -\theta^a \iota_a \big) : \big( r^a \big) & = 0 \\ & = \big( d_{dR} + r^a \iota_{v^a} \big) ( r^a ) \end{aligned}

and for all differential forms $\omega \in \Omega^\bullet\big( X \big)$:

\begin{aligned} \epsilon \circ \big( d_{dR} + d_W\big) \circ : \exp\big( -\theta^a \iota_a \big) : \big( \omega \big) & = \epsilon \circ \big( d_{dR} + d_W\big) \big( \omega - \theta^a \iota_{v^a} \omega + \cdots \big) \\ & = \epsilon \circ \big( d_{dR} \omega + \underset{a}{\sum} \theta^a d_{dR} \iota_{v^a} \omega + \big( r^a - \tfrac{1}{2}f^a_{b c} t^b \wedge t^c \big) \iota_{v^a} \omega + \cdots \big) \\ & = d_{dR} \iota_{v^a} ( \omega) + r^a \iota_{v^a} (\omega) \end{aligned}

because $\epsilon$ annihilates, by (15), all summands containing a $\theta^a$-factor.

The left hand side graded algebra of the isomorphism (13) equipped with the induced differential (16) is called the Cartan model, and that isomorphism exhibits it as equivalent to the Weil model:

(17)$\array{ \Big( \Big( \Omega^\bullet\big(X \big) \otimes \mathbb{R}\big[ \{r^a\}_a \big] \Big)^G \,,\, d_{dR} + r^a \iota_{v^a} \Big) & \underoverset{\simeq}{ : \exp\big( - \theta^a \iota_a \big) : }{\longrightarrow} & \Big( \Big( \Omega^\bullet \big( X \big) \otimes W(\mathfrak{g}) \Big)_{basic} , d_{dR} + d_W \Big) \\ \text{Cartan model} && \text{Weil model} }$

This statement is originally due to Cartan 50, Sec. 6.

In summary, the Cartan model is explicitly the following dgc-algebra:

##### Direct definition

Write

$(\Omega^\bullet(G, \mathfrak{g}^\ast[1])^G \hookrightarrow \Omega(G, \mathfrak{g}^\ast[1])$

for the $G$-invariant differential forms on $G$ with coefficients in the linear dual of the Lie algebra $\mathfrak{g}$, shifted up in degree. So for $\{F^a\}$ a dual basis, a general element of this space in degree $2 p + q$ is of the form

$\omega \;=\; F^{a_1} \wedge \cdots F^{a_p} \wedge \omega_{a_1,\cdots ,a_q} \,,$

where $\omega_{\cdots}$ are differential q-forms, such that for each $t_a \in \mathfrak{g}$ the Lie derivative of these forms satisfies

$\mathcal{L}_{v^a} \omega_{a_1, a_2 \cdots , a_p} = f_{a a_1}{}^b \omega_{b , a_2, \cdots , a_p} + f_{a a_2}^{}^b \omega_{a_1 , b, \cdots , a_p} + \cdots \,,$

where $\{f_{a b}{}^b\}$ are the structure constants of $\mathfrak{g}$ (2).

Equip this graded vector space $\Omega^\bullet(G, \mathfrak{h}^\ast[1])^G$ with a differential $d$ by

$d \colon \omega \mapsto d_{dR}\omega - F^a \iota_{v^a} \omega$

(e.g. Kalkman 93 (1.15)).

The resulting dgc-algebra $(\Omega^\bullet(G,\mathfrak{g}^\ast[1])^G, d)$ is the Cartan model for $G$-equivariant de Rham cohomology on $X$.

### Equivariant de Rham theorem

The point of the above dgc-algebra models is that, under suitable conditions, their cochain cohomology computes the real cohomology of the homotopy type of the homotopy quotient $X \sslash H$, which, as an actual topological space, may be presented by the Borel construction $X \times_G E G$, hence the Borel equivariant de Rham cohomology of $X$.

This is the equivariant cohomology-generalization of the plain de Rham theorem:

###### Proposition

(equivariant de Rham theorem)

Let

1. $G$ be a Lie group which is

2. $X$ be a smooth $G$-manifold (Def. ).

Then the cochain cohomology of (the cochain complex underlying) the Weil model dgc-algebra (11), and hence, by Lemma , also of the Cartan model dgc-algebra (17). is isomorphic to the real cohomology of the homotopy quotient $X \!\sslash\! G$ of the action on (the topological space underlying) $X$ by the (topological group underlying) $G$, hence in particular of the Borel construction $X \times_G E G \simeq X \!\sslash\! G$:

$\array{ \text{Cartan model cohomology} \\ H^\bullet \Big( \Big( \Omega^\bullet\big(X \big) \otimes \mathbb{R}\big[ \{r^a\}_a \big] \Big)^G \,,\, d_{dR} + r^a \iota_{v^a} \Big) \\ {}^{\simeq} \Big\downarrow {}^{ H^\bullet\big( : \exp\big( - \theta^a \iota_a \big) : \big) } \\ H^\bullet \Big( \Big( \Omega^\bullet \big( X \big) \otimes W(\mathfrak{g}) \Big)_{basic} , d_{dR} + d_W \Big) &\underoverset{\simeq}{\;\;\;\;\;\;\;\;\;\;\;\;\;}{\longrightarrow}& H^\bullet \big( X \!\!\sslash\!\! G \,,\, \mathbb{R} \big) \\ \text{Weil model cohomology} && \mathclap{ \text{equivariant real cohomology} } }$
###### Proof idea

Recall that the product topological space $X \times E G$ of $X$ with the total space $E G$ of the universal principal bundle, equipped with the diagonal action by the group $G$, constitutes a resolution of $X$ as a topological G-space, in that the projection

$X \times E G \underoverset{\simeq_{whe}}{ \;\;\; pr_1 \;\;\; }{\longrightarrow} X$

is a $G$-equivariant function which is a weak homotopy equivalence (since $E G$ is a weakly contractible topological space) and the diagonal $G$-action on $X \times E G$ is free (since the action on $E G$ is). Therefore the homotopy quotient of $X$ by $G$ is presented by the ordinary quotient space of $X \times E G$ by $G$, which is what is called the Borel construction

$X \times_G E G \;\coloneqq\; \big( X \times E G \big)/G \;\simeq_{whe}\; X \sslash G$

The point now is that the Weil model (11) for equivariant cohomology is exactly the analog of the Borel construction in terms of dgc-algebraic rational homotopy theory-type models in real cohomology:

By the ordinary de Rham theorem the image of the smooth manifold $X$ in dgc-algebra rational homotopy theory (with real number-coefficients) is given by the de Rham algebra $\Omega^\bullet(X)$, and the image of $E G$ is the Weil algebra $W(\mathfrak{g})$: The contractability of $E G$ corresponds to the free propery (here) of the Weil algebra, and the $G$-action on $E G$ corresponds to the canonical $\mathfrak{g}$-Cartan calculus on $W(\mathfrak{g})$.

Since for a free action the invariant forms are the basic differential forms, this shows that/how the Weil model is the image of the Borel construction in dgc-algebraic rational homotopy theory:

$\,$

###### Remark

A generalization of the equivariant de Rham theorem to non-compact Lie groups exists (Getzler 94) but this uses the simplicial de Rham complex of the action groupoid $X \sslash G$ (Bott-Shulman-Stasheff 76) and is thus a fair bit more complicated, computationally.

### Cartan’s map

If the G-manifold $X$ has a free action, hence is the total space $X = P$ of a $G$-principal bundle $P \to B$, then the Cartan map (or Cartan's map or similar) is a quasi-isomorphism from the Cartan model for the equivariant de Rham cohomology of $X = P$ to the ordinary de Rham complex model for the ordinary de Rham cohomology of the base manifold $B$

$\array{ \bigg( \Big( \Omega^\bullet\big(P \big) \otimes \mathbb{R}\big[ \{r^a\}_a \big] \Big)^G \,,\, d_{dR} + r^a \iota_{v^a} \bigg) & \overset{ \simeq_{qi} }{\longrightarrow} & \Big( \Omega^\bullet_{dR}(B), \, d_{dR} \Big) \\ \omega \otimes \langle - \rangle & \mapsto & \omega_{hor} \otimes \langle F_\nabla\rangle }$

given by choosing an Ehresmann connection $\nabla$ on $P \to B$ and inserting its curvature form into the invariant polynomials $\langle-\rangle$ (essentially the Chern-Weil homomorphism).

## References

The Cartan model for equivariant de Rham cohomology is originally due to

• Henri Cartan, La transgression dans un groupe de Lie et dans un espace fibré principal, Colloque de topologie (espaces fibrés). Bruxelles, 1950

Review:

Early discussion of the Weil model includes

The slick proof of the equivalence between the Weil model and the the Cartan model via the Mathai-Quillen isomorphism (Lemma ) is due to

A review of the Weil model and the Cartan model and the introduction of the “BRST model” (Kalkman model) is in

Generalization of the equivariant de Rham theorem to non-compact Lie groups is due to

based on the simplicial de Rham complex

Discussion of equivariant de Rham cohomology with emphasis on characteristic forms and ordinary equivariant differential cohomology:

Review with emphasis on equivariant localization formulas:

Discussion in relation to resolution of singularities:

Some related discussion for equivariant Riemannian geometry in

• Peter Michor, Basic Differential Forms for Actions of Lie Groups, Proceedings of the American Mathematical Society Vol. 124, No. 5 (May, 1996), pp. 1633-1642 (jstor:)

Discussion in the broader context of equivariant ordinary differential cohomology is in

Discussion in the context of the gauged WZW model includes

Discussion in view of supersymmetry:

Discussion in relation to equivariant K-theory and equivariant elliptic cohomology:

Generalization to equivariant-twisted de Rham cohomology (as the codomain for the twisted equivariant Chern character):

Last revised on September 7, 2021 at 05:35:40. See the history of this page for a list of all contributions to it.